1. Technical Field to which the Invention Pertains
The present invention relates to the rendering techniques which traces the paths of curved light rays and renders the object. More particularly, the present invention relates to the apparatus and method for rendering using symplectic ray tracing which solves the Hamilton's canonical equation by applying symplectic numerical method and traces the paths of light rays.
2. Description of the Related Art
It is a basic technique to visualize objects for analyzing dynamical systems. There are many methods of visualizing dynamical systems by using computer graphics. Among the many methods of visualization, ray tracing is probably one of the most famous methods using computer graphics.
Conventional ray tracing employs straight lines to calculate the paths of light rays. The equation of a straight line is given here.r=a+sv  (Equation1)where r is a location component on the paths of light rays, a is a starting point, v is a direction vector and s is a parameter.
(Equation 1) can be rewritten as follows for the components of the vectors:xi=ai+svi  (Equation2)where xi is a component of vector r. By differentiating twice, (Equation2) can be rewritten as a form of differential equation:                                                         ⅆ              2                        ⁢                          x              i                                            ⅆ                          s              2                                      =        0                            (                  Equation          ⁢                                          ⁢          3                )            
Traditional ray tracing is thus a rendering technique in which the path of a light ray is calculated by solving (Equation 3).
But, there are many non-linear optical phenomena such as a quantity of air above the ground which is a phenomena in non-homogeneously transparent object. People would sometimes like to visualizing the four-dimensional black hole space time.
For example, in non-homogeneously transparent object, the equation for light rays is given by                                                                         ⅆ                n                                            ⅆ                s                                      ⁢                                          ⅆ                                  x                  i                                                            ⅆ                s                                              +                      n            ⁢                                                            ⅆ                  2                                ⁢                                  x                  i                                                            ⅆ                                  s                  2                                                                    =                              ∂            n                                ∂                          x              i                                                          (                  Equation          ⁢                                          ⁢          4                )            where n is the refractive index. (Equation 4) is derived from the Fermat's principle in geometric optics. When n is constant, (Equation 4) reduces to (Equation 3).
In black-hole space time, the equation for a light ray is given by                                                                         ⅆ                2                            ⁢                              x                i                                                    ⅆ                              s                2                                              +                                    Γ                              k                ⁢                                                                  ⁢                l                            i                        ⁢                                          ⅆ                                  x                  k                                                            ⅆ                s                                      ⁢                                          ⅆ                                  x                  l                                                            ⅆ                s                                                    =        0                            (                  Equation          ⁢                                          ⁢          5                )            where Γkli, called the Christoffel symbol, is a function used to calculate the curvature of space. (Equation 5) is known as the geodesic equation.
Traditional ray tracing can not visualize the phenomena in non-homogeneously transparent object or the four-dimensional black hole space time.